In shallow turbulent flows such as floods and tsunami vertical mixing tends to smooth out the flow characteristics in cross-sectional direction. The evolution of the average cross-flow characteristics presents considerable interest. We model such flows using the $k-\omega$ model of turbulence in the framework of the centre manifold theory. We tested the approach on an artificial diffusion problem for which an exact analytical solution is derived. Then we apply the method to model the turbulent flows and deduced the evolution equations for the average velocity, turbulent energy and its rate of dissipation.